Are marriages between say Southern Baptists valid in the eyes of God?

If they had children and they converted to the Catholic Faith would the children be legitimate?

Would they have to be remarried in the Catholic Church?

I have heard that you don't need a priest to get married.

What if one person is Protestant, and the other is unbaptized?  Can that marriage be dissolved if the Protestant becomes Catholic?

Yes.  Neither party is, strictly speaking, Christian, and therefore the Pauline privilege applies but ONLY IF the non-Christian (i.e., the non-Catholic) party refuses the spouse that has converted.  

Remember that the Pauline privilege is the privilege to enter into a second marriage under very strict circuмstances.  If the non-Christian party accepts the conversion of the spouse and is willing to continue the marriage, there is no privilige.

Quote from: TKGS

Yes.  Neither party is, strictly speaking, Christian, and therefore the Pauline privilege applies but ONLY IF the non-Christian (i.e., the non-Catholic) party refuses the spouse that has converted.

The sacramentally valid marriage of two baptized people cannot be dissolved by the Pauline privilege.  It has to be dissolved by the Petrine privilege, through the Pope.

The Pauline privilege applies to valid marriages among the unbaptized.

From the ecclesiastical decisions that have been already quoted, it is clear that the Church has at least the authority of explaining the Pauline privilege, of limiting and extending it. This would give rise to no difficulties if the Pauline privilege, as expressed in 1 Corinthians 7:15, were an immediate Apostolic ordinance and only mediately Divine, inasmuch as Christ would have granted the power in general in a case of necessity to dissolve in favour of the Faith a marriage contracted in infidelity. For the entire Apostolic power passed to the supreme head of the Church, and as the Apostle could determine fixed rules and conditions for the dissolution of the marriages in question, the pope would have precisely the same authority. Yet on this point there is a diversity of opinion among theologians, and the Church has not settled the dispute. For, even if the privilege as promulgated by St. Paul was of immediate Divine right, the Church's power to make at least modifications in case of necessity can readily be explained because such a power belongs to her without a doubt in the other matters that are of Divine right. The first opinion seems to have been held in the fourteenth century by eminent scholars like P. de Palude and de Tudeschis, and in the fifteenth century by St. Antoninus; in recent times it is defended by Gasparri, Rossi, Fahrner, and others. The second opinion is held by Thomas Sanchez, Benedict XIV, St. Alphonsus, Perrone, Billot, Wernz, and others. The instruction of the Holy Office, 11 July, 1866 (Collectan., n. 1353), calls the privilege a Divine privilege "promulgated by the Apostle". However, in spite of the disagreement in regard to the Pauline privilege, the defenders of both opinions agree that there is another method for the dissolution of the marriage of infidels when one of the parties receives baptism, namely, by papal authority. This power is indeed not admitted by all theologians. Even Lambertini (who later became Pope Benedict XIV) doubted it when he was secretary of the Sacred Congregation of the Council, in the causa Florentina, in the year 1726. But earlier papal decisions, as well as the actual decision in this very case, leave no room for doubt that the popes attribute to themselves this power and act accordingly.